RIVER. 



The velocity may alfo become nothing, by the Hope be- 

 coming fo fmall, that 



3°7 



,/S-±h.log.(S + 4S) 



= o; 



but if r=- is lefs than T Tn> i o-<s-*> 



or than T Vth of an inch to 



an Englifh mile, the water will have fenfible motion. 



In a river, the greateft velocity is at the furface, and in 

 the middle of the ftream ; from which it diminifhes towards 

 the bottom, and the fides, where it is the leaft. It has been 

 found, from experiment, that if, from the fquare root of 

 the velocity in the middle of the ftream, expreffed in inches 

 per fecond, unity be fubtra&ed, the fquare of the remainder 

 is the velocity of the bottom. 



Hence, if v be the velocity in the middle of the ftream, 

 the velocity of the bottom will be expreffed by ( v 'v — 1 )' 

 — v — 2 «/v + '• 



The mean velocity, or that with which, were the whole 

 ftream to move, the difcharge would be the fame as the real 

 difcharge, is equal to half the fum of the greateft and leaft 

 velocities, as computed in the laft propofition. Therefore, 

 if v reprefents the greateft velocity, then will the mean ve- 

 locity = v — *JV +§. 



Suppofe that a river, having a rectangular bed, is in- 

 creafed by the junction of another river equal to itfelf, the 

 declivity remaining the fame ; required the increafe of depth 

 and velocity. Let the breadth of the river = b, the depth 

 before the junction d, and after it « ; and, in like manner, i> 



and i>' the mean velocities before and after ; then 



a + 2 a 



~ b * t>i r r 3°7 1/ R 



= R before, and 7 = R' after, fo v = — , 



fuppofing the breadth of the river to be fuch, that we may 

 reject the fmall quantity fubtrafted from R ; and, in like 



manner, v = 



we have 



3°W_R' , 



then fubftituting for R and R', 



^ x J 



bd 



307 



V 2 d + 4' 



/ bx 



V'» 



and 



b + 2x 



Multiplying thefe into the area of the fections bd, bx, we 

 have the difcharges, was. 



b dv = 



b x •»' = 



307 



307 



bd y bd 



</ {6 + 2 «/)' 



b x */ b x 



and 



V (* + 2 •') 



And fince the laft of thefe is double the former, we 

 obtain 



bx *.f bx 2 bd y bd x' 4</' 



V (b + 2x) ~ ' y (b~+ 2 d)' F J + 2» ~ I + 2 <f 



whence x 3 — ( ; , ) x = ,, 



\b + 2 d/ b + 2 d 



a cubic equation folvable by the formula of Cardan, 

 an example, let b = 10 feet, d = l, then a- 5 — 4* = 

 where x = 1.4882, which is the depth of the increafed 

 river. Hence we have 1.4882 x v 1 = 2 v, and 1.4882 : 2 

 :: <v : vf ; or v is to ■»' as 37 to 50 nearly. 

 When the water ia a river receives a permanent increafe, 



As 



10. 



the depth and the velocity, as in the example above, are 

 the firft that are augmented. The increafe of the velocity 

 increafes the action on the fides and bottom, in confequence 

 of which the width is augmented, and fumetimes alfo, but 

 more rarely, the depth. The velocity is thus diminifhed, 

 till the tenacity of the loil, or the hardneis of the rock, affords 

 a fufficient refiftance to the force of the water ; the bed of 

 the river then changes only by infenfible degrees, and, in the 

 ordinary language of hydraulics, is faid to be permanent, 

 though, in itri&nefs, this epithet is not applicable to the 

 courfe of any river. For more on this fubjeft, fee Du Buat, 

 " Principes d'Hydrauliqaes," in two vols. 3vo. Paris, 1806 ; 

 Boffut's " Hydrodynamiques ;" the article River in the 

 " Encyclopedia Britaunica ;" and Playfair's " Outlines of 

 Natural Philofophy." 



The belt and moil iimple method of meafuring the velo- 

 city of the current of a river or canal, is the following. 

 Take a cylindrical piece of dry, light wood, and of a length 

 fomething lefs than the depth of the water in the river ; 

 round one end of it, let there be fufpended as many fmall 

 weights as may be neceffary to keep up the cylinder in a 

 perpendicular fituation in the water, and in fuch a manner 

 that the other end of it may juft appear above the furface of 

 the water. Fix to the centre of that end which appears 

 above water, a fmall and ftraight rod, precifely in the di- 

 rection of the cylinder's axis ; to the end, that when the in- 

 ftrument is fufpended in the water, the deviations of the rod 

 from a perpendicularity to the furface of it, may indicate 

 which end of the cylinder advances the faiteft, by which 

 may be difcovered the different velocities of the water at 

 different depths ; for if the rod inclines forwards, according 

 to the direction of the current, it is a proof that the furfare 

 of the water has the greateft velocity ; but if it inclines 

 back, it (hews that the fwifteft current is at the bottom ; if 

 it remains perpendicular, it is a fign that the velocities at 

 the furface and bottom are equal. 



This inftrument being placed in the current of a river 01 

 canal, receives all the percufiions of the water throughout 

 the whole depth, and will have an equal velocity with that 

 of the whole current from the furface to the bottom at the 

 place where it is put in, and by that means may be found, 

 both with eafe and exactnefs, the mean velocity of that 

 part of the river for any determinate diftance and time. 



But to obtain the mean velocity of the whole fection of 

 the river, the inftrument muft be put fucceffively both in 

 the middle and towards the fides, becaufe the velocities at 

 thofe places are often very different from each other. 

 Having by this means found the difference of time required 

 for the currents to run over an equal f pace ; or, the different 

 dijlances run over in equal times, the mean propartional of all 

 thefe trials, which is found by dividing the common fum 

 of them all by the number of trials, ivillbe the mean velocity 

 of the river or canal. 



If it be required to find the velocity of the current only 

 at the furface, or at the middle, or at the bottom, a fphere 

 of wood, of fuch a weight as will remain fufpended in 

 equilibrium with the water, at the furface or depth which 

 we want to meafure, will be better for the purpofe than a 

 cylinder, becaufe it is only affected by the water of that 

 fole part of the current where it remains fnfpended. 



It is very eafy to guide both the cylinder and the globe in 

 that part which we want to meafure, by means of two 

 threads of fmall cords, which two perfons muit hold and 

 direct, one on each fide the river ; taking care at the fame 

 time neither to retard nor accelerate the motion of the in- 

 ftrument. 



Having 



